Only sort of bike but... I have a Raspberry Pi that I plug my helmet camera into (Contour Roam) that then automatically copies of the video files on connect (to make room for new) and has a script that sets the clock ready for unplugging. I need to add a button to trigger the script (rather than doing it from my phone) and a status light (off=camera not connected, flashing=files being copied, solid=copy done, rapid flashing=clock set/unplug now).
You could make your own custom bike computer? I can't think of something useful that you can't do with a Garmin though.